Im thinking about real-entire functions , strictly increasing on the reals such that

1) they have no real fixpoint , nor at +\- oo. ( -oo is fixpoint of exp(z) + z , but x^2 does not have + oo as fixpoint although oo^2 = oo ; x^2 is not asymp to id(x) ).

2) in the univalent zone near the real axis where f maps univalent to all of C or C\y ( single value y ) , f has no fixpoint.
Equivalent f^[-1](x) , the branch near the real line , has no fundamental fixpoints.

So the fixpoints of f resp Inv f must lie outside the zone resp branch.
I call them " outside " fixpoints.

This makes me wonder about the super of f.
Clearly the outside fixpoints make the super way different.

For instance no fixpioints at + oo i like sexp has.

Reminds me a bit of secondary fixpoint methods.

Not sure what the most intresting f would be.

I assume the simplest topology for the zone comes first. ; no holes for instance.